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Delta-Hedging for Portfolio Insurance
In theory, delta-hedging takes place with computer-
Portfolio insurance, the high-tech hedging strategy that
timed precision, and there aren™t any snags. But in real
helped grease the slide in the 1987 stock market crash,
life, it doesn™t always work so smoothly. “When volatility
is alive and well.
ends up being much greater than anticipated, you
And just as in 1987, it doesn™t always work out as
can™t get your delta trades off at the right points,” says
planned, as some financial institutions found out in the
an executive at one big derivatives dealer.
recent European bond market turmoil.
How does this happen? Take the relatively simple
Banks, securities firms, and other big traders rely
case of dealers who sell “call” options on long-term
heavily on portfolio insurance to contain their potential
Treasury bonds. Such options give buyers the right to
losses when they buy and sell options. But since port-
buy bonds at a fixed price over a specific time period.
folio insurance got a bad name after it backfired on in-
And compared with buying bonds outright, these op-
vestors in 1987, it goes by an alias these days”the
tions are much more sensitive to market moves.
sexier, Star Trek moniker of “delta-hedging.”
Because selling the calls made those dealers vulner-
Whatever you call it, the recent turmoil in European
able to a rally, they delta-hedged by buying bonds. As
bond markets taught some practitioners”including
bond prices turned south [and option deltas fell], the
banks and securities firms that were hedging op-
dealers shed their hedges by selling bonds, adding to
tions sales to hedge funds and other investors”the
the selling orgy. The plunging markets forced them to
same painful lessons of earlier portfolio insurers: Delta-
sell at lower prices than expected, causing unexpected
hedging can break down in volatile markets, just when
losses on their hedges.
it is needed most.
How you delta-hedge depends on the bets you™re
trying to hedge. For instance, delta-hedging would
Source: Abridged from Barbara Donnelly Granito, “Delta-Hedging:
prompt options sellers to sell into falling markets and
The New Name in Portfolio Insurance,” The Wall Street Journal,
buy into rallies. It would give the opposite directions to March 17, 1994, p. C1. Reprinted by permission of Dow Jones &
options buyers, such as dealers who might hold big op- Company, Inc. via Copyright Clearance Center, Inc., © 1994 Dow
tions inventories. Jones & Company, Inc. All Rights Reserved Worldwide.

selling equities based on the put option™s delta, insurers will sell an equivalent number of
futures contracts.4
Several portfolio insurers suffered great setbacks during the market “crash” of October 19,
1987, when the Dow Jones Industrial Average fell more than 20%. A description of what
happened then should help you appreciate the complexities of applying a seemingly straight-
forward hedging concept.
1. Market volatility at the crash was much greater than ever encountered before. Put option
deltas computed from historical experience were too low; insurers underhedged, held too
much equity, and suffered excessive losses.
2. Prices moved so fast that insurers could not keep up with the necessary rebalancing. They
were “chasing deltas” that kept getting away from them. The futures market saw a “gap”
opening, where the opening price was nearly 10% below the previous day™s close. The
price dropped before insurers could update their hedge ratios.
3. Execution problems were severe. First, current market prices were unavailable, with trade
execution and the price quotation system hours behind, which made computation of
correct hedge ratios impossible. Moreover, trading in stocks and stock futures ceased
during some periods. The continuous rebalancing capability that is essential for a viable
insurance program vanished during the precipitous market collapse.

Notice, however, that the use of index futures reintroduces the problem of tracking error between the portfolio and
the market index.

Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

556 Part FIVE Derivative Markets

4. Futures prices traded at steep discounts to their proper levels compared to reported stock
prices, thereby making the sale of futures (as a proxy for equity sales) to increase hedging
seem expensive. While you will see in the next chapter that stock index futures prices
normally exceed the value of the stock index, Figure 15.9 shows that on October 19,
futures sold far below the stock index level. When some insurers gambled that the futures
price would recover to its usual premium over the stock index and chose to defer sales,
they remained underhedged. As the market fell farther, their portfolios experienced
substantial losses.
While most observers believe that the portfolio insurance industry will never recover from
the market crash, the nearby box points out that delta hedging is still alive and well on Wall
Street. Dynamic hedges are widely used by large firms to hedge potential losses from the op-
tions they write. The article also points out, however, that these traders are increasingly aware
of the practical difficulties in implementing dynamic hedges in very volatile markets.

There have been an enormous number of empirical tests of the Black-Scholes option-pricing
model. For the most part, the results of the studies have been positive in that the Black-Scholes
model generates option values quite close to the actual prices at which options trade. At the
same time, some smaller, but regular empirical failures of the model have been noted. For ex-
ample, Geske and Roll (1984) have argued that these empirical results can be attributed to the
failure of the Black-Scholes model to account for the possible early exercise of American calls
on stocks that pay dividends. They show that the theoretical bias induced by this failure cor-
responds closely to the actual “mispricing” observed empirically.
Whaley (1982) examines the performance of the Black-Scholes formula relative to that
of more complicated option formulas that allow for early exercise. His findings indicate that
formulas that allow for the possibility of early exercise do better at pricing than the Black-
Scholes formula. The Black-Scholes formula seems to perform worst for options on stocks
with high dividend payouts. The true American call option formula, on the other hand, seems
to fare equally well in the prediction of option prices on stocks with high or low dividend

F I G U R E 15.9 10
S&P 500 cash-to-
futures spread in 0
points at 15-minute
Source: From The Wall Street
Journal. Reprinted by
permission of Dow Jones &
Company, Inc. via Copyright
Clearance Center, Inc.
© 1987 Dow Jones &
Company, Inc. All Rights
Reserved Worldwide. “40
10 11 12 1 2 3 4 10 11 12 1 2 3 4
October 19 October 20
NOTE: Trading in futures contracts halted between 12:15 and 1:05.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

15 Option Valuation

Rubinstein (1994) points out that the performance of the Black-Scholes model has deteri-
orated in recent years in the sense that options on the same stock with the same expiration
date, which should have the same implied volatility, actually exhibit progressively different
implied volatilities as strike prices vary. He attributes this to an increasing fear of another
market crash like that experienced in 1987, and he notes that, consistent with this hypothesis,
out-of-the-money put options are priced higher (that is, with higher implied volatilities) than
other puts.

• Option values may be viewed as the sum of intrinsic value plus time or “volatility” value.
The volatility value is the right to choose not to exercise if the stock price moves against
the holder. Thus, option holders cannot lose more than the cost of the option regardless of
stock price performance.
• Call options are more valuable when the exercise price is lower, when the stock price is
higher, when the interest rate is higher, when the time to maturity is greater, when the
stock™s volatility is greater, and when dividends are lower.
• Options may be priced relative to the underlying stock price using a simple two-period,
two-state pricing model. As the number of periods increases, the model can approximate
more realistic stock price distributions. The Black-Scholes formula may be seen as a
limiting case of the binomial option model, as the holding period is divided into
progressively smaller subperiods.
• The put-call parity theorem relates the prices of put and call options. If the relationship is
violated, arbitrage opportunities will result. Specifically, the relationship that must be
satisfied is
P C S0 PV(X) PV(dividends)
where X is the exercise price of both the call and the put options, and PV(X) is the present
value of the claim to X dollars to be paid at the expiration date of the options.
• The hedge ratio is the number of shares of stock required to hedge the price risk involved
in writing one option. Hedge ratios are near zero for deep out-of-the-money call options
and approach 1.0 for deep in-the-money calls.
• Although hedge ratios are less than 1.0, call options have elasticities greater than 1.0. The
rate of return on a call (as opposed to the dollar return) responds more than one-for-one
with stock price movements.
• Portfolio insurance can be obtained by purchasing a protective put option on an equity
position. When the appropriate put is not traded, portfolio insurance entails a dynamic
hedge strategy where a fraction of the equity portfolio equal to the desired put option™s
delta is sold, with proceeds placed in risk-free securities.

binomial model, 539 dynamic hedging, 554 option elasticity, 551
Black-Scholes pricing hedge ratio, 550 portfolio insurance, 552
formula, 540 implied volatility, 543 put-call parity
delta, 550 intrinsic value, 532 relationship, 548

1. We showed in the text that the value of a call option increases with the volatility of the
stock. Is this also true of put option values? Use the put-call parity relationship as well as
a numerical example to prove your answer.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

558 Part FIVE Derivative Markets

2. In each of the following questions, you are asked to compare two options with
parameters as given. The risk-free interest rate for all cases should be assumed to be 6%.
Assume the stocks on which these options are written pay no dividends.

a. Price of
Put T X Option
A 0.5 50 0.20 10
B 0.5 50 0.25 10

Which put option is written on the stock with the lower price?
(1) A
(2) B
(3) Not enough information

b. Price of
Put T X Option
A 0.5 50 0.2 10
B 0.5 50 0.2 12

Which put option must be written on the stock with the lower price?
(1) A
(2) B
(3) Not enough information

c. Price of
Call S X Option
A 50 50 0.20 12
B 55 50 0.20 10

Which call option must have the lower time to maturity?
(1) A
(2) B
(3) Not enough information

d. Price of
Call T X S Option
A 0.5 50 55 10

B 0.5 50 55 12

Which call option is written on the stock with higher volatility?
(1) A
(2) B


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